Bernhard Jakoby, Roman Beigelbeck, Thomas Voglhuber-Brunnmaier
The static shielding properties of Faraday wire cages are intuitive as for a small mesh size, the effect on the field can be expected to approach that of an ideal Faraday cage, i.e., a closed conductive surface. However, as it has been recently pointed out, the shielding efficiency is somewhat worse than one might expect and does not particularly conform to the simple approximation of an exponentially decaying field, as it is, e.g., described in The Feynman Lectures on Physics. In the present contribution, we use the case of a circular 2D wire cage to illustrate how the residual field inside such a cage can be visualized in terms of a spatial spectral consideration of the induced charge. It is shown how the residual field in the cage’s center is related to a single Fourier coefficient of this spectral expansion, and that the approximation of the induced charge as a sampled version of the induced charge of a corresponding ideal Faraday cage yields useful approximations for the residual fields close to the cage boundary. The latter also turn out to justify the exponential decay approximation, at least in this region.
The first description of the screening of electric (and in particular, electrostatic) fields by metallic, or more generally, conductive, cages is generally attributed to Faraday’s observations, as reported in his seminal monograph [1]. Today, Faraday cages and their effect can be considered common knowledge, being taught in schools and known to the common public, particularly in connection with protection from lightning strikes, as illustrated by demonstrations in science museums or the protective effect of sitting in a car during a thunderstorm. In 2016, Rajeev Bansal reported in IEEE Antennas and Propagation Magazine’s “Turnstile” column [2] that Chapman et al. investigated the shielding efficiency of Faraday cages composed of grids devising and using rigorous methods, finding that the shielding is not as powerful as may be intuitively expected [3]. For a 2D circular cage, an external line charge in a distance d from the center of the cage creates a remaining field in the center of the cage, which is found to be approximately proportional to $\log(\text{r}/{r}_{0})/\text{nd},$ where $\text{r}/{r}_{0}$ denotes the radius of the wires the cage is made of (normalized to some reference value ${r}_{0})$, and n is the number of wires forming the 2D cage. They also pointed out that apparently, the shielding efficiency before had scarcely been analyzed in a rigorous mathematical manner; at least such an analysis cannot be found in common textbooks or accessible journals, which was the stimulus for their own work. In particular, an (approximately) exponential decay of the fields inside the cage, which is commonly assumed, was proven wrong. As an example, for such a simplified treatment, The Feynman Lectures on Physics [4] are cited stating the origin of the flawed treatment is that “Feynman considers equal charges rather than equal potentials” on the wires. Later in this article, we discuss the context of Feynman’s treatment, illustrating that Feynman’s conclusions can indeed be justified for large or infinitely extended wire grids as has also been pointed out by other authors [5].
The original analysis provided in [3] considered the electrostatic case, which was later extended to the dynamic case [6], where in both cases the devised concept of homogenized boundary conditions was successfully employed. In particular, it is confirmed that the static case can be a useful approximation if the wavelength is large compared to the spacing of the grid (unless spurious resonances are excited in the cage). It is also pointed out that the treatment of lightning and sparking involves complex nonlinear effects such as ionization, which is also not covered by an electrostatic approach. The interested reader seeking a thorough treatment of the problem is particularly referred to these works.
In this contribution, we do not aim at such a rigorous analysis but rather at developing an illustrative view on the problem by utilizing a tool well-known to electrical engineers, i.e., Fourier analysis.
As in [3], for the sake of simplicity, we consider the case of a circular case in two dimensions and treat the purely electrostatic case. This consideration, to a certain extent, also particularly applies to the dynamic cases where the wavelengths involved are much larger than the spacing of the considered grid. For the sake of completeness, we also note that shielding of static and quasi-static magnetic fields is a different (and quite challenging) topic, see, e.g., [7].
In the following, we first introduce a mathematical description of the problem, where the field is described in terms of a Fourier series expansion of the induced charges. We particularly consider the residual field in the center of the cage and close to the cage boundary, where the approximate assumption that the charges in the grid wires represent a sampled version of the charge induced in an ideal (closed) Faraday cage is particularly useful. These considerations are illustrated by means of an example.
We consider 2D electrostatics in free space, i.e., in 3D space everything is considered uniformly extended to infinity with respect to one spatial dimension, say z, such that we can simplify the 3D equations by setting $\partial{/}\partial{z} = {0}{.}$ In the following, we use polar coordinates with radial and angular coordinates r and ${\alpha},$ respectively, where the coordinate center is located in the center of the cage. Figure 1 shows a Faraday cage composed of ${N} = {12}$ circular conductive wires with diameters (widths) w that are uniformly distributed on a circle ${r} = {R},$ i.e., with radius R. In the general case, the wires shall be considered located at angles ${\alpha}_{n} = {n}{2}{\pi}{/}{N}$ with ${n} = {0}\ldots{N}{-}{1}{.}$ We consider the important case of thin wires, referring to the assumption that the diameter w of each wire is significantly smaller than the spacing between them and the dimension of the cage, which essentially means that ${w}\ll{2}{\pi}{R}{/}{N}{.}$ As in [3], we consider the field generated by a 2D point charge (which is a line charge in 3D) located in a distance d from the cage’s center and lying on the x-axis as excitation in our problem. Even though we consider static fields, in this article we use the term incident field, commonly used in electromagnetic scattering, to address this field contribution.
Figure 1. A 2D circular wire cage and a line charge in some distance d, serving as a source for the external field to be screened.
In electrostatics, the electric field is conservative and the field vector E can be expressed as the negative gradient of the electric potential $\varphi,$ i.e., ${E} = {-}\nabla\varphi{.}$ In charge-free regions, such as the region within the cage, the potential $\varphi$ fulfills Laplace’s equation $\Delta\varphi = {0}{.}$ When using polar coordinates, the potential can be written as a function $\varphi{(}{r},{\alpha}{)}$ and will be ${2}{\pi} {-} {\text{periodic}}$ in ${\alpha}$ such that it can be expanded in a Fourier series with respect to ${\alpha}{:}$ (Note that two types of indices are used in this article. Whenever an index refers to a spectral Fourier component, we use the letter k, whereas n is used as the index for the cage wires ${n} = {0}\ldots{N}{-}{1}{.)}$
Here the imaginary unit is denoted by ${j} = \sqrt{{-}{1}}{.}$ The r-dependent Fourier coefficient ${\tilde{\varphi}}_{n}(\text{r})$ is thereby given by \[{\tilde{\varphi}}_{k}{(}{r}{)} = \frac{1}{{2}{\pi}}\mathop{\int}\nolimits_{0}\nolimits^{{2}{\pi}}{\varphi{(}{r},{\alpha}{)}\exp{(}{-}{jk}{\alpha}{)}{d}{\alpha}} \tag{2} \] which is related to the coefficients of the alternative sine and cosine representations by \begin{align*}{\tilde{\varphi}}_{\text{c},\text{k}} & = {\tilde{\varphi}}_{k} + {\tilde{\varphi}}_{{-}{k}}\,{(}{\text{for}}\,{k}\geq{0}{)} \\ {\tilde{\varphi}}_{\text{s},\text{k}} & = {j}\left({{\tilde{\varphi}}_{k}{-}{\tilde{\varphi}}_{{-}{k}}}\right)\,{(}{\text{for}}\,{k}\geq{1}{)}{.} \tag{3} \end{align*}
Inserting the expansion (1) into Laplace’s equation $\Delta\varphi = {0}$ in polar coordinates leads to a simple general solution for the r-dependent Fourier coefficients ${\tilde{\varphi}}_{k}(\text{r}),$ or alternatively, ${\tilde{\varphi}}_{\text{c},\text{k}}(\text{r})$ and ${\tilde{\varphi}}_{\text{s},\text{k}}(\text{r}),$ which, inside the cage, yields an r-dependence proportional to ${r}^{\mid{k}\mid}$ [see “Supplement A” of the supplementary material (available at 10.1109/MAP.2022.3229287), where the case ${k} = {0}$ is also discussed]. Each spectral coefficient corresponds to a particular contribution to the total field.
This already gives a clue as to what we need to identify in the remaining field in the cage’s center. Close to the center, i.e., for vanishing r, out of all the terms proportional to ${r}^{|\text{k}|},$ the asymptotically dominant one will be proportional to r, i.e., the one for ${k} = \pm{1}{.}$ Thus, the associated spectral coefficients essentially determine the remaining field close to the center.
Close to the cage, i.e., when r approaches R, the situation is not as clear and depends on amplitude distribution of the individual spectral coefficients.
The remaining fields close to the cage boundary as well as those close to the center can serve as particular measures for the screening efficiency of the cage. As outlined in the next section, considering the field in the cage’s center is in line with the conclusions made in [3], whereas the fields just behind the wire grid happen to support Feynman’s conclusion [4] to a certain extent, as is further discussed.
To discuss all of this, we move on to look into the utilization of the Fourier series expansion in a bit more detail.
The field remaining inside the cage can be represented as a superposition of the incident field and the field generated by the induced charges on the wires of the cage, which ideally cancel out the former as well as possible. To determine the residual field, the first step is to solve for the induced charges, and there are a variety of analytical and numerical as well as approximate methods to obtain these. Later in this section, we show some simple approaches for illustrational purposes but will not particularly stress this point as there are many suitable ways for calculating electrostatic fields, which are extensively discussed in the literature. For the treated circular Faraday cage, we particularly refer to [3], which provides a more profound insight into these topics.
For the present, let us assume that the charge distribution in the cage wires is already known, and we consider how the residual field can be obtained. The charge distribution on each wire is located at the wire’s surface and, in general, is not uniformly distributed across the surface. This is in contrast to the simple case of a single isolated charged wire with a circular cross section where the surface charge would be uniformly distributed. In this case, the field generated outside of the wire by these surface charges is identical to that of an equivalent line charge located at the wire’s center (replacing the wire). This is a consequence of the equivalence principle of electrostatics (see, e.g., [8]). Similarly, even though in our case the charge on a particular wire’s surface will not be uniformly distributed, in distances sufficiently large compared to the wire’s diameter w, the field originating from the charges on a particular wire will approach that of an equivalent line charge ${q}_{n}$ (for wire number n) placed instead of the wire at its center location at the angle ${\alpha}_{n} = {n}{2}{\pi}{/}{N},$ which can be used to approximately evaluate the field within the cage.
These equivalent line charges, which are located at positions ${r} = {R}$ and ${\alpha}_{n} = {n}{2}{\pi}{/}{N}$ (with ${n} = {0}\ldots{N}{-}{1}{),}$ can, in turn, be conceptually described in terms of a surface charge distribution ${\sigma}{(}{\alpha}{)}$ located at the circle (cylinder in 3D) ${r} = {R}$ \[{\sigma}{(}{\alpha}{)} = \frac{1}{R}\mathop{\sum}\limits_{{n} = {0}}\limits^{{N}{-}{1}}{{q}_{n}}{\delta}\left({{\alpha}{-}{n}\frac{{2}{\pi}}{N}}\right) \tag{4} \] using Dirac delta functions ${\delta}{(}{\alpha}{)}$ to represent the line charges, where ${q}_{n}$ are the associated line charge densities. The factor $1/\text{R}$ maintains proper scaling, i.e., integrating the surface charge density around the entire circumference yields the sum of the line charges: \[\mathop{\int}\nolimits_{0}\nolimits^{{2}{\pi}}{{\sigma}{(}{\alpha}{)}{Rd}{\alpha} = \mathop{\sum}\limits_{{n} = {0}}\limits^{{N}{-}{1}}{{q}_{n}}{.}} \tag{5} \]
This representation nicely connects to the case of the ideal Faraday cage, where a distributed surface charge distribution on the circle ${r} = {R}$ is obtained. As this surface charge exactly compensates the incident field inside the circle, it can also be used to represent the incident field in terms of an equivalent charge distribution, which is simply equal to the negative ideal surface charge density.
As discussed in “Supplement A” of the supplementary material (available at 10.1109/MAP.2022.3229287), a Fourier series representation allows for a convenient and simple representation of fields and charges. Also, the surface charge representing the equivalent line charges from (4) can be expanded into a Fourier series yielding \[{\tilde{\sigma}}_{k} = \frac{1}{{2}{\pi}{R}}\mathop{\sum}\limits_{{n} = {0}}\limits^{{N}{-}{1}}{{q}_{n}}\exp\left({{-}{jnk}\frac{{2}{\pi}}{N}}\right) \tag{6} \] as Fourier series coefficients, which are periodic in the discrete variable k. One period, e.g., from ${k} = {0}$ to ${k} = {N}{-}{1},$ corresponds to the spectral representation of the discrete Fourier transform (DFT), which can be efficiently computed using the fast Fourier transform (FFT) algorithm. (Note that a periodic but continuous function corresponds to a discrete spectrum, as described by the Fourier series. A periodic and discrete function corresponds to a periodic and discrete spectrum, as described by the DFT. The FFT is a numerically efficient way to calculate the DFT, which is why these terms are sometimes used alternately.) Although the spatial charge distribution in terms of line charge is discrete, the fields generated by these line charges are, of course, continuous, and to calculate their spectral representation we stick to the classical Fourier series.
In the following, we consider that the external field exciting the cage is provided by a line charge in a distance d from the cage’s center, as shown in Figure 1, which yields symmetric fields and charges with respect to ${\alpha}{.}$ In doing so, the real-valued representation of the Fourier series (1) is particularly handy as the coefficients of the sine terms vanish, e.g., ${\tilde{\varphi}}_{\text{s},\text{k}} = {0}$ for the potential. Also, using the resulting cosine series, we have to deal with only positive spectral indices k, which eases the notation in the following considerations, yet they similarly apply to asymmetric cases. For this special case, the Fourier cosine coefficients, e.g., of the surface charge density, are related to the general coefficients by \[{\tilde{\sigma}}_{\text{c},\text{k}} = {2}{\tilde{\sigma}}_{k} = {2}{\tilde{\sigma}}_{{-}{k}}{(}{k}\geq{0}{)}{.} \tag{7} \]
Under these assumptions, analogous to (1), we therefore have \[{\sigma}{(}{r},{\alpha}{)} = \frac{{\tilde{\sigma}}_{\text{c},0}}{2} + \mathop{\sum}\limits_{{k} = {1}}\limits^{\infty}{{\tilde{\sigma}}_{\text{c},\text{k}}}{(}{r}{)}\cos{(}{k}{\alpha}{)}{.} \tag{8} \]
The zeroth-order coefficient ${\tilde{\sigma}}_{\text{c},0}$ corresponds to the total charge of the cage, which has to vanish in the case of electrostatic induction because the wires are connected and initially uncharged. When calculating the charges numerically, a nonvanishing, zeroth-order coefficient can thus be an indicator of inaccuracies. As outlined in “Supplement A” of supplementary material (available at 10.1109/MAP.2022.3229287), the potential ${\varphi}_{s}$ due to the charges on the wire can be similarly expanded in a Fourier series with respect to ${\alpha}{.}$ (Note that the subscript “s” stands for “scattered,” indicating that this field is related to the induced charges as opposed to the “incident” field generated by the exciting charge.)
As a result, a simple solution ansatz for the r-dependent coefficients is obtained, i.e., \begin{align*}{\tilde{\varphi}}_{\text{s},\text{c},\text{k}}{(}{r}{)} & = {A}_{k}{r}^{k}\,\,{\text{inside the cage}}\,{(}{r}\,{<}\,{R}{),}\,{\text{and}} \\ {\tilde{\varphi}}_{\text{s},\text{c},\text{k}}{(}{r}{)} & = {B}_{k}{r}^{{-}{k}}\,\,{\text{outside of the cage}}\,{(}{r}\,{>}\,{R}{)} \tag{9} \end{align*} with yet-to-be-determined coefficients ${A}_{k}$ and ${B}_{k}{.}$ By virtue of the orthogonality of the expansion functions in the Fourier series (cosine functions in the present case), the spatial domain interface conditions across sheets of surface charge translate into corresponding conditions for every spectral component. In particular, by virtue of Gauss’s law, the surface-normal component of the electric displacement (in free space ${\epsilon}_{0}\text{E})$ experiences a jump when passing through a sheet carrying a surface charge, which yields the following condition for the spectral coefficients of potential and charge at the surface ${r} = {R}{:}$ \[{\left.{\frac{\partial{\tilde{\varphi}}_{s,c,k}{(}{r}{)}}{\partial{r}}}\right|}_{{r} = {R} + }{-}{\left.{\frac{\partial{\tilde{\varphi}}_{s,c,k}{(}{r}{)}}{\partial{r}}}\right|}_{{r} = {R}{-}} = \frac{{\tilde{\sigma}}_{\text{c},\text{k}}}{{\epsilon}_{0}}{.} \tag{10} \]
Together with continuity of the potential at ${r} = {R},$ i.e., \[{\left.{{\tilde{\varphi}}_{s,c,k}}\right|}_{{r} = {R} + } = {\left.{{\tilde{\varphi}}_{s,c,k}}\right|}_{{r} = {R}{-}} \tag{11} \] one can solve for the coefficients ${A}_{k}$ and ${B}_{k},$ and the following solution of the Fourier coefficient for the potential inside the cage is obtained: \[{\tilde{\varphi}}_{\text{s},\text{c},\text{k}}{(}{r}{)} = {\left({\frac{r}{R}}\right)}^{k}\frac{R}{2\text{k}{\epsilon}_{0}}{\tilde{\sigma}}_{\text{c},\text{k}} \tag{12} \] and therefore, by inserting these coefficients in the Fourier cosine series expansion, we obtain the desired representation of the field in terms of the generating charges \[{\varphi}_{s}{(}{r},{\alpha}{)} = \mathop{\sum}\limits_{{k} = {1}}\limits^{\infty}{{\left({\frac{r}{R}}\right)}^{k}}\frac{R}{2\text{k}{\epsilon}_{0}}{\tilde{\sigma}}_{\text{c},\text{k}}\cos{(}{k}{\alpha}{)}{.} \tag{13} \]
It is particularly useful that this approach can not only be used to represent the field generated by the induced charges (the “scattered” fields) but also the incident field. As the charge density ${\sigma}_{\text{id}}$ induced in an ideal circular Faraday cage exactly cancels out the incident field inside the cage and as ${\sigma}_{\text{id}}$ represents a distribution of surface charges on the surface ${r} = {R},$ the incident field can be expressed in terms of these charges and their Fourier series coefficients. “Supplement B” of the supplementary material (available at 10.1109/MAP.2022.3229287) summarizes some fundamental properties for the ideal circular Faraday cage and also provides a closed-form expression for Fourier coefficients of the induced surface charge density ${\tilde{\sigma}}_{\text{id},\text{l},\text{n}}$ for the special case when the incident field stems from a line charge, as depicted in Figure 1 [see (28)].
The answer to this question depends on the criterion used to evaluate it. It is near at hand to start with considering the field at the center of the cage as it can be expected that the remaining field will be smaller close to the center. But what can also be of interest is how fast the field decreases right behind the screen when moving toward the center. Let us have a look at these cases.
The electric field at the center of the cage ${(}{r} = {0}{)}$ can be fully characterized in terms of its radial component [note that at ${r} = {0},$ we have the following simple relation between the polar components of the electric field ${E}_{r}{(}{0},{\alpha}{)} = {E}_{\alpha}{(}{0},{\alpha}{-}{\pi}{/}{2}{),}$ and thus, all the information is contained in one of these components]: \[{\left.{{E}_{r}}\right|}_{{r} = {0}} = {-}\frac{\partial\varphi}{\partial{r}} = {-}\frac{\partial{\varphi}_{s}}{\partial{r}}{-}{E}_{\text{inc},\text{r}}{.} \tag{14} \]
Here, ${\varphi}_{s}$ denotes the electric potential generated by the induced charges on the wire cage, and ${E}_{\text{inc},\text{r}}$ denotes the r component of the electric field generated by external charges, i.e., the line charge at ${x} = {d}$ (and ${y} = {0}{)}$ in our example. ${E}_{\text{inc},\text{r}}$ is given simply by the field of a line charge in distance d and readily obtained as \[{E}_{\text{inc},\text{r}} = \frac{{-}{q}_{\text{inc}}\cos{\alpha}}{2{{\pi}{\epsilon}}_{0}\text{d}}{.} \tag{15} \]
Utilizing the Fourier series expansion (13) for the scattered field, the derivative with respect to r can be taken for every element of the series (representing a power series with respect to r), and upon setting ${r} = {0},$ we are left with a single nonzero element of the series, i.e., ${k} = {1},$ yielding for the total field \[{E}_{r}{(}{0},{\alpha}{)} = {-}\frac{{\tilde{\sigma}}_{\text{c},1}}{2{\epsilon}_{0}}\cos{\alpha}{-}\frac{{q}_{\text{inc}}}{2{{\pi}{\epsilon}}_{0}\text{d}}\cos{\alpha}{.} \tag{16} \]
So the remaining field in the center is essentially determined by the first-order Fourier series coefficient ${\tilde{\sigma}}_{\text{c},1}$ of the induced charge distribution! This equation is exact, i.e., not an approximation, and holds for an arbitrary charge distribution along the circle ${r} = {R}{;}$ it also holds for the continuous charge distribution induced in an ideal Faraday cage, which is considered in “Supplement B” of the supplementary material available at 10.1109/MAP.2022.3229287. As can be seen in (28), the corresponding Fourier coefficient for ${k} = {1}$ indeed exactly cancels the incident field as it should be as the charges induced in the ideal cage cancel the incident field everywhere inside the cage ${(}{r}{<}{R}{)}{.}$ [Note that (28) in “Supplement B” of the supplementary material (available at 10.1109/MAP.2022.3229287) provides the two-sided Fourier coefficient, which has to be multiplied by two to obtain the corresponding coefficient of the cosine series.]
Again, we can use the spectral Fourier representation for the induced charges to establish the residual field. At first sight, it is near at hand to assume that the induced charges in the wire cage will closely resemble a “sampled” version of the continuous charge distribution of the ideal Faraday cage ${\sigma}_{\text{id}}{(}{\alpha}{)}$ [see “Supplement B” of the supplementary material (available at 10.1109/MAP.2022.3229287)]. Indeed, as illustrated in this section and in the example, this turns out to be a useful approximation close to the cage boundary, however, its accuracy deteriorates when the residual fields in the cage’s center (i.e., for ${r}\ll{R}{)}$ are calculated. Figure 2 illustrates the idea of sampling and the resulting spectra. The continuous charge distribution (upper-left plot), which, by definition, is periodic in ${\alpha},$ corresponds to a spectrum of discrete Fourier coefficients. The charge distribution can be “sampled,” yielding a set of N line charges uniformly distributed at angles ${\alpha}_{n} = {n}{2}{\pi}{/}{N}$ around the circumference ${r} = {R}{.}$ In the sampled version, the strength of each line charge is proportional to the value of the continuous charge distribution sampled at the location of the respective line charge, i.e., at the corresponding angle ${\alpha}_{n}$ (lower-left plot). As it is well known from the spectral analysis of sampled signals (see, e.g., [9]), the spectrum, i.e., the Fourier coefficients representing this arrangement of line charges, is given by a periodic repetition of the coefficients associated with the sampled (continuous) function. The coefficients contained within one spectral period can also be obtained by applying an FFT to the original charge coefficients.
Figure 2. Spectra (Fourier series coefficients) of continuous and sampled charge distribution. The spectrum associated with the sampled distribution is a periodic repetition of the original one.
The line charge densities ${q}_{n}$ associated with the sampled version in (4) are given by \[{q}_{n}\approx\frac{{2}{\pi}{R}}{N}{\sigma}_{\text{id}}\left({n\frac{{2}{\pi}}{N}}\right){.} \tag{17} \]
The factor ${2}{\pi}{R}{/}{N}$ represents the length of one of N equally sized segments of the circle and relates the surface charge density at the sampling location to an associated (approximate) line charge density. Obviously, for sufficiently large N, the compensating field generated by such a distribution of line charges can be expected to come arbitrarily close to the continuous charge distribution on the ideal Faraday cage, and also, the sampled version cancels out the incident field fairly well. However, particularly for small N, the sampled ideal charge distribution does not represent the actually induced charges very accurately, as also pointed out in [3]. Thus, the actually induced charge distribution on the wires tends to compensate the incident external field not as well as it would be possible, in principle, for an arrangement of line charges. Yet, on second thought, the physical requirement determining the charge distribution among the wires is not the cancellation of the incident field inside the cage but the boundary condition of a constant potential at the wire surfaces, which obviously represents a different requirement.
Still, approximation of the sampled ideal charge distribution provides some very valuable insights and turns out to be useful, particularly when considering fields close to the cage boundary, as illustrated further in this section. Considering the corresponding discrete charge distribution as given in (17) according to the spectral properties of sampled signals (see, e.g., [9]), the resulting Fourier coefficients are \[{\tilde{\sigma}}_{k}\approx\mathop{\sum}\limits_{{m} = {-}\infty}\limits^{ + \infty}{{\tilde{\sigma}}_{{\text{id}},{k}{-}{mN}}} \tag{18} \] i.e., the spectrum for the sampled version is given by the sum of the periodically repeated spectra of the underlying continuous function ${\sigma}_{\text{id}}$ with a period N, as presented in Figure 3. Note that here we use the complex Fourier series (rather than the cosine series), which allows for representing the impact of sampling in a simpler fashion.
Figure 3. (a) The spectrum of the charge distribution induced in the ideal Faraday cage, (b) the sampled ideal distribution, which may serve as an, albeit coarse, approximation of the actually induced charges, and (c) the spectral sum of the sampled distribution plus the equivalent charge distribution ${\tilde{\sigma}}_{\text{inc}}$ generating the incident field inside the cage. Note that the Fourier coefficient for ${k} = {0}$ vanishes due to the charge neutrality, i.e., the total charge on the cylinder vanishes. The spectrum of the sampled charge distribution is given in terms of a periodic repetition of the spectrum of the function being sampled. As can be seen, when adding the distributions ${\tilde{\sigma}}_{\text{i}\text{d},\text{s}\text{a}\text{m}\text{p}\text{l}\text{e}}$ and ${\tilde{\sigma}}_{\text{inc}},$ the “baseband” contribution cancels out such that the residual field is entirely given in terms of the shifted spectra in this approximation.
The spectrum is now not only discrete but also periodic and, as mentioned previously, the coefficients within one period ${(}{n} = {0}\ldots{N}{-}{1}{)}$ can be efficiently obtained using the FFT algorithm, taking the FFT of the discrete charges ${q}_{n}$ in spatial domain. Depending on the width of the ideal charge distribution’s spectrum ${\tilde{\sigma}}_{\text{id},\text{k}},$ the shifted spectra will, in general, overlap to a certain extent (see also the discussion later in this section). If we now want to obtain the field generated by the induced charges, we employ (13). [We note that (13) refers to the Fourier cosine expansion; for symmetric situations, these are related to the complex series coefficients by (7).] Note that even though the charge spectrum is periodic, the spectrum of the potential is, in general, not periodic as the potential is a continuous function of ${\alpha}$ in spatial domain. At the same time, due to the intrinsic periodicity in spatial ${(}{\alpha}{)}$ domain, the spectrum remains discrete. Hence, the associated spectral coefficients of the potential for a given r are also not periodic, which is represented by the fact that the kth series coefficient in (13) features a factor $(\text{r}/\text{R}{)}^{k}/\text{k},$ which is obviously not periodic in k.
To represent the incident field inside the cage, we now use the previously mentioned fact that, inside the cage, the incident field can be represented simply as the field that would be created inside the cage by a fictitious charge distribution ${\sigma}_{\text{inc}}$ on ${r} = {R},$ which is equal to the negative ideal charge distribution, i.e., ${\sigma}_{\text{inc}}{(}{\alpha}{)} = {-}{\sigma}_{\text{id}}{(}{\alpha}{)}{.}$ This is the case as ${\sigma}_{\text{id}}$ would exactly compensate for the incident field inside the cage.
If we now adopt our preliminary assumption that the induced charges on the wires are approximately equal to a sampled version of the ideal continuous distribution ${\sigma}_{\text{id}}{(}{\alpha}{),}$ the total field inside the cage is approximately related to an “effective” charge distribution whose spectrum is that of the periodically repeated spectrum of the ideal distribution (representing the sampled ideal charge distribution) as given by (18) but omitting the fundamental period ${m} = {0},$ which is cancelled out by adding the negative ideal charge distribution ${\sigma}_{\text{inc}}{(}{\alpha}{)} = {-}{\sigma}_{\text{id}}{(}{\alpha}{),}$ which effectively cancels the “baseband” of the periodically repeated spectrum. This is illustrated in the bottom plot of Figure 3(c). Therefore, using the “sampling approximation,” the residual field is related to the entire spectrum of the sampled charge distribution, except for the “baseband” contribution ${(}{k} = {0}{)}$ in the series in (18).
This effective charge distribution ${\tilde{\sigma}}_{k} + {\tilde{\sigma}}_{\text{inc},\text{k}}$ can now be used to calculate the residual field using (13). As previously pointed out, this series for the potential inside the cage ${(}{r}{<}{R}{)}$ features coefficients proportional $(\text{r}/\text{R}{)}^{k}/\text{k}$ which, close to the cage boundary ${r} = {R},$ essentially impose an additional decay, with $1/\text{k}$ for increasing k.
We now introduce a further approximation by assuming that the spectral width of the ideal charge distribution ${\tilde{\sigma}}_{\text{id},\text{k}}$ is small compared to the shift parameter N in (18). This also means that neighboring shifted spectra overlap only moderately (which is also assumed in Figures 3 and 4). This condition will often be fulfilled as a Faraday cage will typically feature a mesh width, within which the external field will only moderately change, which in a spectral domain translates into a narrow spectral width of ${\tilde{\sigma}}_{\text{id},\text{k}}$ compared to the number of samples N, which also means that the sampling is more “dense.” Consider now that apart from constant factors, the kth spectral coefficient of the potential is essentially given by the associated spectral coefficient of the charge times a k-dependent factor $(\text{r}/\text{R}{)}^{k}/\text{k},$ as given in (12). In our current consideration, the relevant charge spectrum (including an effective charge distribution accounting for the incident field) is approximately given the periodically repeated spectrum of the ideal charge distribution where the fundamental period around ${k} = {0}$ is almost entirely canceled out (see Figure 4). The envelope associated with the factor $(\text{r}/\text{R}{)}^{k}/\text{k},$ which has to be applied to obtain the spectrum of the potential, is also indicated as a dashed line in Figure 4. The shifted charge spectra are centered around ${k} = {mN}$ with integer m. If the spectral widths of these shifted spectra are small, which means that they are essentially concentrated around ${k} = {mN},$ instead of the factor k-dependent factor $(\text{r}/\text{R}{)}^{k}/\text{k},$ we can use an approximately constant factor $(\text{r}/\text{R}{)}^{mN}/\text{mN}$ to be applied to each of these partial, shifted spectra. This in turn means that upon transformation back into spatial ${(}{\alpha}{)}$ domain, the mth partial spectrum (shifted by mN) is weighed with a constant factor $(\text{r}/\text{R}{)}^{mN}/\text{mN}$ and, using the Fourier shift theorem for the inverse transform, results in the ideal charge distribution ${\sigma}_{\text{id}}{(}{\alpha}{)}$ times an additional phase term $\exp{(}{jmN}{\alpha}{),}$ accounting for the spectral shift by mN. Combining the terms for $\pm{m},$ the phase terms yield a cosine function $\cos{(}{mN}{\alpha}{)}$ for each partial spectrum such that the entire spectrum is given by \begin{align*}\varphi{(}{r},{\alpha}{)} & \approx{\sigma}_{\text{id}}{(}{\alpha}{)}\frac{R}{{\epsilon}_{0}}\mathop{\sum}\limits_{{m} = {1}}\limits^{\infty}{\frac{{\rho}^{mN}}{mN}}\cos{(}{mN}{\alpha}{)} \\ & = {-}{\sigma}_{\text{id}}{(}{\alpha}{)}\frac{R}{{\epsilon}_{0}}\frac{\ln\left[{{1} + {\rho}^{2N}{-}{2}{\rho}^{N}\cos{(}{N}{\alpha}{)}}\right]}{2\text{N}} \tag{19} \end{align*}
Figure 4. The spectrum of the actually induced charge distribution $\tilde{\sigma}$ plus the equivalent charge distribution ${\tilde{\sigma}}_{\text{inc}}$ representing the incident field. As the real induced charge distribution does not exactly correspond to the sampled ideal charge distribution (as considered in Figure 3), the “baseband” contribution is not entirely canceled out by ${\tilde{\sigma}}_{\text{inc}}{.}$
where ${\rho} = {r}{/}{R}$ and $\ln$ denotes the natural logarithm. Note that the series starts at ${m} = {1}$ instead of zero, which accounts for the canceled baseband contribution. The closed-form solution for the series was obtained by considering a related geometric series for the complex variable ${Z} = {\rho}^{N}\exp{(}{jN}{\alpha}{),}$ where the original series can be represented as the real part of the indefinite integral of the geometric series for Z with respect to Z. This approximation is particularly appealing as it can be established for arbitrary incident fields as the ideal charge distribution ${\sigma}_{\text{id}}{(}{\alpha}{)}$ can be readily expressed in terms of the incident field at the cage boundary [see (26)]. The existence of this closed-form solution shall, however, not cloud the main point, i.e., that the residual field behind the cage boundary (and its decay), by virtue of considering the sampling approximation, can be directly related to the spectral repetitions of the ideal charge distribution.
A simpler, yet less accurate approximation is obtained by keeping only the first term of the series featuring the slowest decay for decreasing ${\rho},$ yielding \[\varphi{(}{r},{\alpha}{)}\approx{\sigma}_{\text{id}}{(}{\alpha}{)}\frac{R}{\text{N}{\epsilon}_{0}}{\rho}^{N}\cos{(}{N}{\alpha}{)}{.} \tag{20} \]
The electric field components can be readily obtained from the potential by \[{E}_{r} = {-}\frac{\partial\varphi}{\partial{r}} \tag{21} \] and \[{E}_{\alpha} = {-}\frac{1}{r}\frac{\partial\varphi}{\partial{\alpha}} \tag{22} \] which shows that the field components show an r-dependence ${r}^{{(}{N}{-}{1}{)}}{.}$ Note again that this approximation is particularly useful to describe the fields close to the screen, while it becomes particularly inaccurate when ${r}\rightarrow{0},$ where it predicts that the field strength vanishes entirely.
As shown in the previous section, the field at the center is essentially determined by the first spectral coefficient of the induced charge [see (16)]. If the charges on the wires were actually sampled versions of the ideal continuous charge distribution, the field at the center would indeed almost completely vanish as the field generated by the “baseband” contributions of the periodically repeated charge spectrum, i.e., the term associated with ${m} = {0}$ in (18), is exactly canceled by the incident field. Referring to (16), for vanishing ${E}_{r}$ at ${r} = {0},$ the first Fourier coefficient of the induced charge density would have to be ${\tilde{\sigma}}_{\text{c},1} = {2}{\tilde{\sigma}}_{1} = {-}{q}_{\text{inc}}{/}{\pi}{d}{.}$ Referring to (28) in “Supplement B” of the supplemental material (available at 10.1109/MAP.2022.3229287), we find that this is exactly the first Fourier coefficient of the charge distribution of the ideal Faraday cage, i.e., ${\tilde{\sigma}}_{\text{id},1}{.}$ Yet, the sampled ideal charge distribution would have additional, albeit small, contributions to ${\tilde{\sigma}}_{1},$ stemming from the shifted spectra, i.e., the terms associated with ${m}\ne{0}$ in (18), which is why the field in the center would not exactly vanish, even if the sampled ideal charge distribution were induced in the cage wires. Hence, as discussed in the previous section, if one wants to know the remaining field in the center of the cage, the first spectral coefficient has to be determined exactly, which is not feasible with the sampling approximation, as will also be illustrated in the examples presented in the next section. When moving close to the cage boundary $\left({{r}\rightarrow{R}}\right),$ however, the contributions of the spectral “baseband” become less prominent. Therefore, the sampled ideal charge approximation and others based thereon can at least qualitatively describe the field behavior. In particular, we can expect a dominating r-dependence $(\text{r}/\text{R}{)}^{N}$ for the potential, as shown in (20).
If we increase the cage size ${R}\rightarrow\infty$ but keep the distance (or arc length) between the wires, i.e., ${a} = {2}{\pi}{R}{/}{N},$ constant, we can introduce a scaled-coordinate ${\xi} = {(}{R}{-}{r}{)/}{a},$ which, for increasing ${\xi}{>}{0},$ leads from the cage boundary to the inside to the cage. The dominant dependence $(\text{r}/\text{R}{)}^{N}$ turns into ${(}{1}{-}{2}{\pi}{\xi}{/}{N}{)}^{N},$ which, in the limit ${N}\rightarrow\infty,$ yields an exponential decay \[\mathop{\lim}\limits_{{N}\rightarrow\infty}{\left({{1}{-}\frac{{2}{\pi}{\xi}}{N}}\right)}^{N} = \exp{(}{-}{2}{\pi}{\xi}{)} \tag{23} \] which corresponds to the statement given in The Feynman Lectures on Physics [4]. This means that Feynman’s approximation is still valid for a large cage if the cage’s wire spacing is small enough for the sampling approximation to be made.
Still, deep inside the cage, the approximation is not as good as there are substantial contributions from the remaining charges in the “baseband” present (see Figure 4), which are small but yield fields, which do not decay as strongly, as will also be illustrated by the examples in the next section. In “Supplement C” of the supplementary material (available at 10.1109/MAP.2022.3229287), relevance of the approximations made is briefly summarized.
In this section, we show the shielding efficiency and application of the different approximations presented for a simple example. As we are not so much interested in the shielding efficiency itself (which has been studied in the references given earlier) but rather demonstrate the degree of validity of the approximations, and particularly, the underlying consideration of periodically repeated spectra associated with the sampling approximation, we refrain from extended parameter studies. All the calculations were performed using scaled quantities, where the scaled line charge density ${\hat{\rho}}_{l}$ is related to the line charge density ${\rho}_{l}$ by ${\hat{\rho}}_{l} = {\rho}_{l}{/(}{2}{{\pi}{\epsilon}}_{0}{)}$ with the vacuum permittivity ${\epsilon}_{0}{.}$ For excitation by a single line charge, we set ${\hat{\rho}}_{l}$ to unity yielding associated units for the resulting fields (labeled as “a.u.” in the plots).
In particular, we consider a cage consisting of ${N} = {30}$ wires, which will be excited by the field of a positive line charge in a distance d from its center (see also Figure 1). Alternatively, we also briefly consider a uniform external field afterward. The radius of the wire cage amounts to ${r} = {d}{/}{2},$ and the diameter of the wires is ${w} = {0}{.}{01}\,{d}{.}$ This means that the wire diameter amounts to roughly 10% of the spacing between the wires. The resulting potential landscape is shown in Figure 5 for the case of an external uniform field (oriented in a –x direction) and for the field generated by the external line charge. In both cases, it can be seen that the potential inside the cage is fairly constant, corresponding to a vanishing electric field. Also, the location of the wires is clearly visible as they pin the potential to a constant value, i.e., the potential of the cage.
Figure 5. The electrostatic potential around a 2D circular wire cage ${(}{N} = {30}$ wires) exposed 1) to a uniform electric field oriented in the −x direction (the radius of the cage is 0.5 in used scaled units) and 2) to the field generated by a line charge at ${x} = {d},$ where the radius of the cage is 0.5 d and the diameter of the individual wires is ${w} = {0}{.}{01}{d}{.}$ These results were obtained using a numerical simulation, as described in “Supplement C” of the supplementary material (available at 10.1109/MAP.2022.3229287). a.u.: arbitrary units.
Solving for the charges on the wire, we compare the resulting Fourier coefficients of the equivalent line charge array with the Fourier coefficients of the induced continuous charge distribution in an ideal Faraday cage, which is shown in Figure 6. As discussed previously in this section, the coefficients ${\tilde{\sigma}}_{k}$ approximately represent a superposition of spectra of ${\tilde{\sigma}}_{\text{id},\kern0.1em\text{k}}$ shifted by $\pm{mN}{.}$
Figure 6. The Fourier coefficients ${\tilde{\sigma}}_{k}$ corresponding to the equivalent line charges (representing the charges on the cage wires) yield a periodic pattern of Fourier coefficients, which can be obtained using the FFT algorithm. The Fourier coefficients of the induced charge distribution ${\tilde{\sigma}}_{\text{id},\text{k}}$ of the corresponding ideal Faraday cage (a conducting cylinder) are similar for low n. The coefficients ${\tilde{\sigma}}_{k}$ approximately correspond to a superposition of periodically repeated patterns of ${\tilde{\sigma}}_{\text{id},\text{n}}$ as the induced charges on the wires are approximately equal to the sampled ideal charge distribution. (The coefficients are all real and negative; the plot shows their absolute value.)
The field around the cage in terms of the potential illustrated in Figure 5 was obtained using the numerical method outlined in “Supplement C” of the supplementary material available at 10.1109/MAP.2022.3229287. In Figure 7, we show the corresponding radial field strength ${E}_{r}$ according to the numerical solution and the approximate closed-form solution corresponding to (19). The field was obtained in polar ${(}{r},{\alpha}{)}$ coordinates and is plotted in 3D surface plots using the polar coordinates as independent axes. The figure qualitatively suggests that the approximation is reasonable. To investigate this in more detail, we plot ${E}_{r}$ versus ${\alpha}$ for three different r: ${r} = {0}{.}{99}\,{R},{r} = {0}{.}{93}\,{R},$ and ${r} = {0}{.}{87}\,{R}{.}$ It can be seen that accuracy of the approximation deteriorates quite quickly by increasing the distance ${R}{-}{r}$ from the cage boundary ${r} = {R},$ yet the magnitude of ${E}_{r}$ also decreases rapidly (consider the y-axis scaling of the three plots) such that this relative error is not shown as clearly in the plot given in Figure 7, i.e., the absolute error is comparatively small everywhere.
Figure 7. (a) The radial field component ${E}_{r}$ inside the cage upon excitation by an external line charge versus the polar coordinates r and ${\alpha}$ and (b) versus ${\alpha}$ for different constant radii r. For the (a) plots, the upper plot gives the numerical solution and the lower plot the approximation according to (19).
This behavior can also be seen when plotting ${E}_{r}$ versus r; we did this for two fixed angles ${\alpha}{:}$ ${\alpha} = {0}$ corresponding to the axis formed by the cage’s center and exciting the line charge (where the strongest field ${E}_{r}$ can be expected), and ${\alpha} = {0}{.}{125}\times{(}{2}{\pi}{)}$ corresponding to a 45° angle to this axis. In the first case, the approximation appears reasonable, but a closer inspection reveals that the residual ${E}_{r}$ at the center of the cage ${(}{r} = {0}{)}$ is not reproduced; here the approximation yields a zero field. For ${\alpha} = {0}{.}{125}\times{(}{2}{\pi}{),}$ the approximation is qualitatively worse but again, the fields are much smaller in absolute value.
Figure 8 also depicts the characteristics for ${\alpha} = {0},$ again with a magnified inset highlighting the behavior of small field strengths at the origin. The closed-form approximation approaches zero while the numerical result yields a small, yet nonzero, residual field. The dashed-dotted line indicates the field in the center according to (16), which corresponds to the first spectral coefficient. The numerical result asymptotically approaches this value for ${r}\rightarrow{0}{.}$ Note that this residual field is related to the difference of the two first ${(}{k} = {1}{)}$ spectral coefficients shown in Figure 6.
Figure 8. The radial field component ${E}_{r}$ inside the cage upon excitation by an external line charge versus r for different constant angles ${\alpha}{.}$ These plots again represent cross sections through the ones given in Figure 7(a). The inset in (a) ${(}{\alpha} = {0}{)}$ shows a zoomed-in version illustrating the (limited) accuracy of the approximation close to the cage’s center, showing the validity of the center field approximation (16) at the same time as indicated by the “Field in Center From First Fourier Coefficient.”
Figures 9–11 in “Supplement D” of the supplemental material (available at 10.1109/MAP.2022.3229287) show similar plots for the angular field component ${E}_{\alpha}$ conforming to the same observations. Note that ${\alpha} = {0}$ was not selected as the scan line for the plot versus r as ${E}_{\alpha}$ vanishes along this line; instead, ${\alpha} = {0}{.}{017}\times{(}{2}{\pi}{)}$ was used, which corresponds to the radial line, which is approximately between the first two wires of the cage. Furthermore, Figures 12–17 in “Supplement D” of the supplemental material (available at 10.1109/MAP.2022.3229287) show corresponding plots for a uniform incident field.
The shielding efficiency of a circular Faraday cage with respect to arbitrary external (“incident”) electrostatic fields was considered as a 2D problem, demonstrating the application of a Fourier approach to represent the remaining field in the cage, where fields and charges were expanded into a Fourier series with respect to the angular coordinate. It turns out that the remaining field in the center of the cage is directly related to the first-order Fourier series coefficient of the charge distribution induced in the cage wires. A simple, approximate consideration where the induced field was expressed as a sampled version of the continuous charge distribution of an equivalent ideal Faraday cage (a conducting cylinder) showed that the field close to the cage boundaries can be expressed in terms of the shifted spectra that were a result of the sampling. Based on this consideration, a simple, closed-form approximation was established. The approach also explained to what extent the often-disputed claim that the field decays exponentially behind the boundary can be maintained.
Roman Beigelbeck is grateful for stimulating discussions with Andreas Kainz and other researchers involved in grant 3619S92411 from the German Federal Office for Radiation Protection. This article has supplementary downloadable material available at 10.1109/MAP.2022.3229287.
Bernhard Jakoby (bernhard.jakoby@jku.at) is a full professor with the Institute for Microelectronics and Microsensors at the Johannes Kepler University, A4040 Linz, Austria. He has worked in academic research in Belgium and the Netherlands as well as in the German automotive industry. He is a member of the Austrian Academy of Sciences and a Fellow of the IEEE.
Roman Beigelbeck (roman.beigelbeck@donau-uni.ac.at) is a senior scientist at the Department for Integrated Sensor Systems, the University for Continuing Education, A2700 Krems, Austria. He has (co)-authored more than 150 peer-reviewed papers. His current research comprises (semi)-analytical modeling techniques for sensing and thin-film characterization applications.
Thomas Voglhuber-Brunnmaier (thomas.voglhuber-brunnmaier@jku.at) is a senior researcher with the Institute for Microelectronics and Microsensors at the Johannes Kepler University Linz, A4040 Linz, Austria. Since 2008, he has been working on the modeling of microsensors for liquid property sensing.
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Digital Object Identifier 10.1109/MAP.2022.3229287